Reviewing the Portfolio Optimization Engine
our white paper"Machine Optimization: A General Framework for Portfolio Selection"A logical framework for portfolio optimization is provided considering certain assumptions about the expected relationship between risk and return. We explore the fundamental roots of common portfolio weighting mechanisms, such as market capitalization and fair weighting, and discuss the rationale for various risk-based optimizations, including minimum variance, maximum diversification, and risk parity.
For each portfolio selection method, we examine the conditions that make mean-variance selection optimal. For example, the market capitalization weight is mean-variance-optimal if the returns are fully explained by the CAPM beta, or in other words, if all investments have the same expected Treynor ratio. A minimum variance weighted portfolio is optimal if all investments have the same expected return, while a maximum diversification weighted portfolio is optimal if the investments have the same Sharpe ratio.
The Portfolio Optimization Machine framework presents how well academic theories about the relationship between risk and return explain what we observe in real life. While academics would have investors believe that riskier investments should generate higher returns, we generally don't see this relationship.
For example, we show that the stock market line, which shows the relationship between returns and the beta of stocks, and the capital market line, which plots the relationship between returns and volatility, are flat orfallFor historical samples of US and international stocks. In other words, stock returns are independent or inversely related to risk.
We also analyze the returns of all major asset classes, including global stocks, bonds and commodities. For asset classes, there appears to be a positive relationship between risk and return, at least when looking at returns in different macroeconomic regimes. Normalizing for inflation and growth environments, stocks and bonds appear to have the same Sharpe indices in the historical sample.
Since 1970, under institutional conditions, the Sharpe ratio of diversified commodities has been about half that of stocks and bonds. However, we emphasize that our analysis may be biased against commodities due to few commodity-friendly regimes in our historical sample. With such a small sample size, we believe it is premature to reject the hypothesis that commodity risk should offset at the same rate as stock and bond risk.
ourWhite paperA great deal of theory is advanced and guidance is provided on the nature of the relationship between risk and return based on history. With this guidance, we can invoke optimization machine decision trees to make informed guesses about the best portfolio options for different investment domains.
We show that optimization machines are useful guides to optimal portfolio formation, but that the relative chances of optimized versus naive methods depend on the size of diversification opportunities relative to the number of assets in the investment universe. We instantiated a new term "Quality Ratio" to measure the amount of any investment field1.
data and methods
In this document, we test the optimized machine framework. Specifically, we use optimization machines to predict which portfolio approaches are theoretically optimal based on our knowledge of the observed historical relationship between risk and return. We then test these predictions by running simulations on various data sets.
We simulate our research on a known piece of paper(DeMiguel, Garlappi y Uppal)2007)Titled "Optimal and Naive Diversification: How Inefficient Is a 1/N Portfolio Strategy?" it discusses some of the major technical issues that complicate the practical use of portfolio optimization. The authors also present the results of empirical tests of various portfolio optimization methods on multiple data sets to compare the performance of optimal versus naive methods.
(DeMiguel, Garlappi y Uppal)2007)Simulate all equity investment domains. To provide potentially more practical information, we also simulate global asset classes whose returns are derived from different sources of risk, such as regional equity indices, global bonds, and commodities.
Specifically, we evaluate the performance of the original and optimized portfolios on the following data sets, which are available every day:
- 10 US Market Cap Weighted Industrial Portfolios from the Ken French Database
- 25 US capitalization-weighted factor equity portfolios, sorted by size and book-to-market (i.e., value) from the Ken French database
- 38 US market capitalization-weighted sub-sector portfolios from the Ken French database
- 49 US market capitalization-weighted sub-sector portfolios from the Ken French database
- 12 global asset classes from multiple sources2
We build portfolios at the end of each quarter, with a one-day lag between calculating optimal portfolio weights and trading.
(DeMiguel, Garlappi y Uppal)2007)A variety of portfolio formation methods were tested, including long, short, and long versions of minimum variance and mean variance optimization. They also tried different types of contraction methods to manage estimation error.
We'll follow a similar process, but we'll place long, one-to-one constraints on all optimizations, and use a sample covariance of 252 days (ie, one business year) without any reduction methods. The length-only constraints are an acknowledgment of the fact that practitioners are aware of the instability of unconstrained optimization. We will discuss reduction methods in a later post when we discuss more robust optimization methods.
For our simulations, we will compare the performance of the naive approach (equally weighted and market capitalization weighted) with portfolios formed using the following optimizations, all with long constraints (w>‹0), the sum of the weights is 1 ($\sum_i^Nw=1$).
Portfolio optimization methods
Please note that all but one of the optimization descriptions below are in ourPortfolio Optimization White Paper, and are repeated here for convenience only. If you are familiar with the specification and optimization equivalence conditions for these optimizations in the whitepaper, we encourage you toJump to the description of Hierarchical Minimum Variance Optimization.
minimum variance
If all investments had the same expected return regardless of risk, investors looking to maximize return with minimal risk should focus on minimizing risk. This is the explicit goal of a minimum variance portfolio.
$$w^{MV}=\operator name*{arg\,min}_{}w^T\cdot\Sigma\cdot w$$
Wherelittleis the covariance matrix.
(Haugen and Baker1991)It is wise to abandon any relationship between risk and reward, at least for stocks. His article was one of the first to show that beta does not explain stock returns well. In fact, they observed a negative relationship between returns and volatility.
Confronted with the spurious link between risk and reward,(Haugen and Baker1991)A long-only minimum variance portfolio that recommends periodic reorganization can dominate a market capitalization-weighted stock portfolio.
maximum diversification
(Choueifaty y CoignardYear 2008)It proposes that markets be efficient against risk so that investments generate returns proportional to their returns.total risk, measured by volatility. This differs from CAPM, which assumes that returns are proportional to non-diversifiable (ie systematic) risk. Choueifaty et al. describe their approach as Maximum Diversification, for reasons that will become clearer later.
Consistent with the notion that returns are proportional to volatility, the maximum diversification optimization substitutes returns for asset volatility in the maximum Sharpe ratio optimization, taking the following form.
$$w^{MD}=\nombre del operador*{arg\,max}_{}\frac{w \times \sigma}{\sqrt{w^T\cdot\Sigma\cdot w}}$$
WherepagylittleSee the volatility vector and the covariance matrix, respectively.
Note that the optimization seeks to maximize the ratio of the weighted average volatility of the portfolio components to the total volatility of the portfolio. This is analogous to maximizing weighted average returns when returns are proportional to volatility.
An interesting implication, which is explored in detail in the follow-up document（Choueifaty、Froidure 和 Reynier2012)It is the ratio maximized in the optimization function that quantifies the degree of diversification of the portfolio. This is very intuitive.
The volatility of a perfectly correlated portfolio will be equal to the weighted sum of the volatilities of its components, since there is no opportunity for diversification. when the asset isSoyWith perfect correlation, the weighted average volatility becomes greater than the portfolio's volatility, proportional to the degree of diversification available.
The diversification-to-maximize ratio quantifies the degree to which portfolio risk can be minimized by strategically placing weights on diversified (not perfectly correlated) assets.
maximum decorrelation
The maximum decorrelation described(Christopherson et al.2010)Closely related to Minimum Variation and Maximum Diversification, but for situations where an investor believes that all assets have similar returns and volatility, but they are not correlated. It is a minimum variance optimization performed on the correlation matrix instead of the covariance matrix. W
Interestingly, when the weights derived from maximal decorrelation optimization are divided by their respective volatilities and re-normalized to sum to 1, we recover the maximum sparse weights. Therefore, portfolio weights that maximize decorrelation will also maximize the diversification ratio when all assets have the same volatility and will maximize the Sharpe ratio when all assets have the same risk and return.
The maximally decorrelated portfolio is found by solving the following problem:
$$w^{MDec}=\operator name*{arg\,min}_{}w^T\cdot A\cdot w$$
where A is the correlation matrix.
risk parity
The minimum variance and maximum diversification portfolios are valid under the intuitive assumption of mean variance. Minimum variance is efficient if the assets have similar returns, while maximum diversification is efficient if the assets have similar Sharpe indices. However, both methods have the disadvantage that they can be highly concentrated in a small number of assets. For example, a minimum variance portfolio would be disproportionately allocated to less volatile assets, while a highly diversified portfolio would focus on assets with high volatility and low market covariance. In fact, these optimizations can result in a portfolio holding only a fraction of all available assets.
In some cases this may not be desirable. Concentrated portfolios may also not be able to accommodate large amounts of capital without high market impact costs. Additionally, concentrated portfolios are more susceptible to misestimates of volatility or correlation.
These problems motivate the search for optimization heuristics that satisfy similar optimization goals but are less focused. Equal-weight and capitalization-weighted portfolios are common examples of this, but there are other approaches that are persuasive under different assumptions.
Inverse Volatility and Inverse Variance
When investments have similar expected Sharpe ratios and investors cannot reliably estimate the correlations (or we can assume that the correlations are homogeneous), optimal portfolios are weighted by the inverse of the asset's volatility. Inverse-variance portfolios are medium-variance optimal when the investments have similar expected returns (regardless of volatility) and unknown correlations. Note that when all investments have the same pairwise correlations, the inverse volatility portfolio is consistent with the most diversified portfolio, while the inverse variance portfolio approximates the minimum variance portfolio.
The weights for the inverse volatility and inverse variance portfolios are found by:
$$w^{IV}=\frac{1/\sigma}{\sum_{i=1}^{n}1/\sigma}$$
$$w^{IVar}=\frac{1/\sigma^2}{\sum_{i=1}^{n}1/\sigma^2}$$
Wherepagis a vector of asset volatility, andpag2is a vector of asset variations.
contribution of equal risk
(Maillard, Roncalli and TelecheYear 2008)An optimization of equal contribution to risk is described, which is fulfilled when all the assets contribute the same volatility to the portfolio. Equal risk contribution portfolios have been shown to be a compelling balance between equal weight and minimum variance portfolio objectives. It is also a close cousin of the inverse volatility portfolio, except that it is less susceptible to situations where the assets have very different correlations.
The weights of the equal risk contribution portfolios are found following the convex optimization with the formula(spin2013):
$$w^{ERC}=\nombre del operador*{arg\,min}_{}\frac{1}{2}w^T\cdot\Sigma\cdot w – \frac{1}{n}\sum_{我=1}^n\ln(w_i)$$
Wherelittleis the covariance matrix.
An equal contribution to risk portfolio will have positive weights on all assets, and is optimal in terms of mean variance when all assets are expected to contribute the same marginal Sharpe ratios (relative to the equal contribution portfolio itself). risk). Therefore, optimal matching is based on the assumption that portfolios that contribute the same risk are macroeconomically efficient. It has been shown that the portfolio will have a volatility between the minimum variance portfolio and the equal weight portfolio.
Hierarchical Minimum Variance
(Lopez Prado2016)proposes a novel approach to portfolio construction, which he calls "hierarchical risk parity." The stated purpose of this new approach is to "solve the three main problems of general quadratic optimizers and the Markowitz CLA.3In particular: unsteadiness, difficulty concentrating and poor performance. "
In addition to the known sensitivity of mean variance optimization to errors in mean estimates, De Prado also recognized that traditional optimizers are also brittle because they require matrix inversion and the role of determinants, which can arise when the matrix is not a well-qualified question. A matrix with a high condition number is numerically unstable and can lead to an undesirably high load on economically insignificant factors.
(Lopez Prado2016)Assertion-related structures contain ordinal information that can be exploited by organizing assets into hierarchies. The goal of hierarchical risk parity is to change/restructure the covariance matrix to be as close to the diagonal matrix as possible without changing the covariance estimates. The minimum variance combination of a diagonal matrix is the inverse variance combination. For this reason, we describe the method as stratified minimum variance. We explain many of these concepts in more detail in follow-up articles.4.
review(DeMiguel, Garlappi y Uppal)2007)
We now turn to the results of an article, "Optimal Diversification vs. Naive Diversification: How Inefficient Is the 1/N Portfolio Strategy?" After(DeMiguel, Garlappi y Uppal)2007), which is often cited to refute optimization-based methods. According to the paper, the authors are motivated by a desire to "understand the conditions under which mean-variance optimal portfolio models perform well even in the presence of estimated risk." the use of the 1/N heuristic as an asset allocation strategy, but rather as a benchmark for evaluating the performance of various portfolio rules proposed in the literature".
While we are committed to re-examining the(DeMiguel, Garlappi y Uppal)2007), we question the generalizability of this paper for several important reasons. First, the authors choose to specify their model in a way that, while technically accurate, violates most common-sense portfolio management practices. Furthermore, they opted for an empirical analysis of a universe designed almost perfectly to confound optimization-based approaches.
confusing methodology
The specification problem relates primarily to the way the authors measure the optimized mean and covariance. For example, they run simulations to build optimal portfolios each month based on rolling estimation windows of 60 and 120 months.
This is strange for several reasons. First, the authors cite no evidence that investors use these estimation windows in practice to form optimal portfolios. Therefore, there is no reason to believe that your method represents a meaningful optimization use case.
Second, the authors provide no evidence or theory why the estimates for the 60- and 120-month windows should be informative of next month's returns. In fact, they did an analysis of stock portfolios and there is evidence that stock portfolios are mean reverting over the long term.
The main case for the existence of long-run mean reversion is presented in two papers published in 1988, one of which was presented by(Potterba y Summers)1988), another for(Fame and French1988)The conclusion of these articles is that for periods of 3 to 5 years (ie, 36 to 60 months), there is a long-term mean reversion in stock market returns between 1926 and 1985. Three-year returns years have a 25% negative correlation, while five-year returns have a 40% negative correlation.
If the returns of the last 5 to 10 years are(DeMiguel, Garlappi y Uppal)2007)To estimate the portfolio mean, we must expect the optimal portfolio to disappoint, because portfolio optimization will predict above-average returns during periods when it actually produces below-average returns, and vice versa.
The authors also emphasize that the grand universe scarcity problem confounds estimates of covariance. Specifically, if the estimation window length is less than the dimensionality of the matrix, the covariance matrix will be ill conditioned. To find the optimal weights for 500 values, you need at least 500 data points for each value. At monthly granularity, this requires 42 years of data, while we need 10 years of weekly data.
However, the test dataset used in the paper is also available at daily granularity. The covariance matrix is adequately conditioned and optimized daily for 500 values with less than two years of data. One wonders why the authors use data on a monthly basis when daily data is available.
Homogeneous investment fields
The investment domain used to compare the performance of naive versus optimal diversification approaches seems ill-chosen, as the authors' stated goal is to "understand the conditions under which mean-variance optimal portfolio models can be expected to perform." good". The authors analyze an investment universe that consists entirely of stock portfolios. Clearly, stock portfolios are dominated by a single source of risk, stock beta, and offer few opportunities for diversification.
(DeMiguel, Garlappi y Uppal)2007)Acknowledge the problem directly in the article:
... the 1/N rule works well for the data sets we are considering [because] we are using it to distribute wealthbriefcaseactions instead of individual actions. Because diversified portfolios have lower idiosyncratic volatility than individual assets, naive diversification loses much less than optimal diversification by spreading wealth across portfolios. Our simulations show that optimal diversification policies dominate the 1/N rule [emphasis ours] only at very high levels of idiosyncratic volatility.
Idiosyncratic volatility is simply the volatility of the residuals after regression of asset returns on the dominant systematic risk factors. For equity portfolios, for example, portfolios of surveyed sectors, industries and factors(DeMiguel, Garlappi y Uppal)2007), these are the residuals of the stock betas. If most of the variance in the test universe is explained by stock betas, then idiosyncratic volatility will be very small, as will the possibility of diversification.
Few diversification opportunities
One way to determine the amount of idiosyncratic risk within a range of assets is to use principal component analysis (PCA). PCA is a tool for identifying potentially independent (ie, uncorrelated) sources of risk, or principal components, of an investment.
The result of PCA is the eigenvalues,Raise, which describes the total amount of variance explained by each principal component, and the eigenvectorA, which describes the sensitivity or "beta" of each asset to each principal component. There are always the same number of eigenvalues and eigenvectors as there are inversions, so a set of 10 inversions will decompose into 10 eigenvectors with associated eigenvalues.
Order the main components so that the first componentRaise1is the one that explains the greatest variance. For the stock domain, the first principal component is considered to represent the beta coefficient of the market. Therefore, the first eigenvalue quantifies the amount of total portfolio variance explained by the market beta.
All other principal components represent the direction of risk regardless of market beta. Therefore, the total amount of heterogeneous variance in the asset domain is equal to1‰−Raise1.5
We examine the amount of idiosyncratic risk available to provide diversification for each of the domains we surveyed in Figure 1. We also show a breakdown of the most diverse areas of the major futures markets to highlight diversification opportunities beyond asset classes. traditional assets. The graph shows the amountHeterogeneousDiversification risk, so lower bars mean fewer diversification opportunities.
Figure 1:Specific risks of the different investment fields.
Source: ReSolve Asset Management calculations. Industry and portfolio data from Ken French's database sorted by size and market capitalization. Country stock index data from Global Financial Data. Asset class data for the S&P Dow Jones Indices. CSI futures data. The risk of heterogeneity is calculated as 1: the proportion of the total variance explained by the first principal component.
You can see that about three-quarters of the variance in the industry universes and factor ordering is explained by the first principal component, which represents the beta of US stocks. Only one-quarter of the Risks are idiosyncratic and can be used to enhance diversification.
By contrast, around two-thirds and four-fifths of the asset class and futures space, respectively, come from sources other than the first principal component. This leaves more idiosyncratic variation for optimization methods to take advantage of diversification opportunities.
Number of independent "bets"
Understand how small the opportunities for diversification are(DeMiguel, Garlappi y Uppal)2007)When selecting investment areas, we find it useful to quantify the number of uncorrelated sources of return (ie independent bets) available in each investment group.
remember that(Choueifaty y CoignardYear 2008)shows that the diversification ratio of a portfolio is the ratio between the weighted sum of the volatility of assets and the volatility of the portfolio after accounting for diversification.
$$DR=\frac{\sum_i^nw_i\sigma_i}{\sigma_P}$$
This is intuitive because if all the assets in the portfolio were correlated, the weighted sum of their volatilities would equal the portfolio volatility and the diversification index would be 1. As the assets become less correlated, the portfolio volatility will decrease due to diversification, while the weighted sum of component stock volatilities will remain the same, causing the ratio to increase. When all assets are uncorrelated (pairwise correlation is zero), each asset in the portfolio represents an independent bet.
Consider 10 assets with homogeneous pairwise correlations. Figure 2 shows how the number of independent bets available decreases as the pairwise correlation increases from 0 to 1. Note that when the correlation is 0, there are 10 bets, as each asset responds to its own source of risk. . When the correlation is 1, there is only 1 bet, since all the assets are explained by the same source of risk.
Figure 2:The number of independent bets is represented as an equally weighted portfolio of 10 assets with equal volatility based on the pairwise average correlation.
Source: ReSolve Asset Management. For illustrative purposes only.
（Choueifaty、Froidure 和 Reynier2012)Show that the number of independent risk factors within a range of assets is equal to the square of the diversification index of the most diversified portfolio.
Going one step further, we can find the number of independent (ie uncorrelated) risk factors that are ultimately available across the spectrum of assets by first solving for the weights that satisfy the most diversified portfolio. We then take the square of that portfolio's diversification index to get the number of unique risk directions if we maximize diversification opportunities.
We apply this approach to count the number of independent sources of risk available to investors in each of our test universes. Using the full data set available for each universe, we solve for the maximum diversified portfolio weights and square the diversification index. We find that 10 industry portfolios, 25 factor portfolios, 38 sub-industry portfolios, and 49 sub-industry portfolios yield 1.4, 1.9, 2.9, and 3.7 unique sources of risk, respectively. The results are summarized in Figure 3.
These are pretty amazing results. In 10 industry portfolios and 25 factor portfolios, fewer than 2 uncorrelated risk factors were at play. When we expand to 36 and 49 subsectors, we get fewer than 3 and 4 factors, respectively.
To put this in perspective, we also counted the number of independent factors at play in our test range of 12 asset classes and found 5 independent bets. Going one step further, we also analyzed stand-alone bets on stock indices, bonds and commodities across 48 major futures markets and found 13.4 uncorrelated risk factors.
image 3:The number of independent risk factors that exist in the investment universe.
Source: ReSolve Asset Management calculations. Industry and portfolio data from Ken French's database sorted by size and market capitalization. Country stock index data from Global Financial Data. Asset class data for the S&P Dow Jones Indices. CSI futures data. The number of independent bets is equal to the square of the diversification index of the most diversified portfolio formed using pairwise perfect correlations across the entire data set.
Formulate a hypothesis from an optimization machine.
The optimization machine was created to help investors choose the most suitable optimization for any investment field based on the attributes of the investment and the investor's beliefs. Specifically, an optimized machine decision tree guides investors to the portfolio formation method most likely to generate an optimal medium-variance portfolio, given some or all of the volatility, correlation, and/or positive perceptions of the overall relationship, if any.
One of the most important qualities for an investor to consider is the degree of diversification available relative to the number of assets. If the amount of diversification available is small relative to the number of assets, the noise in the covariance matrix can dominate the signal.
We refer to the relationship between the number of independent bets and the number of assets in the investment universe as "Quality Ratio". "Quality Ratio" is a good indicator of the amount of "Signal to Noise Ratio" diversification in the investing world. When the quality index is high, we expect the optimized methods to dominate the naive methods. When it is low, investors should expect that using more sophisticated techniques will produce only a very small boost in risk-adjusted return.
For example, the overall quality index for the 10 industry portfolio is 0.12, while the overall quality index for the 49 sub-industries is 0.08. Compare to our overall asset class quality ratio of 0.42.
Figure 4:Quality Ratio: Number of Independent Bets / Number of Actives.
Source: ReSolve Asset Management calculations. Industry and portfolio data from Ken French's database sorted by size and market capitalization. Country stock index data from Global Financial Data. Asset class data for the S&P Dow Jones Indices. CSI futures data. The number of independent bets is equal to the square of the diversification index of the most diversified portfolio formed using pairwise perfect correlations across the entire data set. The quality ratio is the number of independent bets / the number of assets.
The mass ratio helps inform expectations about how well optimized methods can compete overall with naive methods. For universes with low mass ratios, we expect the naive approach to dominate optimization, while universes with relatively high mass ratios can benefit from optimal diversification.
In cases where high-quality indices drive investors to opt for optimization, the next step is to choose the optimization method that is most likely to achieve mean variance efficiency. An optimization decision tree is a useful guide because it raises questions about what portfolio parameters can be estimated and the expected relationship between risk and return. The answers to these questions lead directly to an appropriate approach to portfolio formation.
Optimizing most branches of the decision tree leads to heuristic optimization, which avoids the need to estimate individual asset returns by expressing returns as functions of different forms of risk. For example, maximum diversification optimization expresses the view that returns are proportional to volatility and linearly proportional, while minimum variance optimization expresses the view that investments have the same expected return regardless of risk. Therefore, these optimizations do not require any mean estimates, only volatility or covariance estimates.
which is good because(Chopra and Ziemba1993)Show that the optimization is much more sensitive to errors of the sample mean than to errors of volatility or covariance. The authors show that for investors with a relatively high risk tolerance, the impact of the mean estimation error is 22 times greater than that of the covariate estimation error. For investors with lower risk tolerance, the relative impact of the error in the sample mean rises to 56 times the error in the covariance. This does not mean that investors should always shy away from optimizations that take a positive view of returns; rather, investors should take steps to minimize the general terms of error. We will discuss this concept in detail in a future article.
For now, we will limit our optimization options to common risk-based methods such as minimum variance, maximum diversification, and risk parity. Optimization is useful if we assume that we cannot gain any advantage through better performance estimates. Later, we will explore how to incorporate systematic active views, such as those presented by popular factor strategies such as momentum, value, and trend.
Let's use an optimization engine to infer which portfolio formation method should generate the best results for each investment domain. Specifically, we want to predict which optimization method is most likely to produce the highest Sharpe ratio.
Whichever optimization is chosen, the magnitude of the optimization relative to the equally weighted return will depend largely on the quality index of the investment universe. Sector and factor stock portfolios have lower quality indices, which should yield marginal improvements compared to an equal weight approach. Asset class domains have higher quality indices, suggesting that we should see more significant optimization performance relative to equal weights.
Stock Market Sector and Factorial Portfolios
Remember our role"Machine Optimization: A General Framework for Portfolio Selection"Historically, stock returns have been unrelated or negatively correlated with beta and volatility. Any risk-based optimization is based on a positive or zero relationship between risk and return, since an inverse relationship violates the fundamental principles of financial economics (particularly rational utility theory), so we will assume stocks of industries and factors. Portfolio returns are all equal and independent of risk.
From the portfolio optimization decision tree, we see that an equal-weighted portfolio is mean-variance optimal if the assets have the same expected return and have the same volatility and correlation. A minimum variance portfolio is also medium variance optimal if the assets have the same expected return, but the optimization also takes into account differences in expected volatility and heterogeneous correlations.
Ex ante minimum variance portfolios should outperform equally weighted portfolios if the covariances are heterogeneous (i.e., unequal) and the observed covariance in our estimation window (252-day rolling returns) is There are reasonably good estimates of the covariance of the period (in our case a calendar quarter).
Consider which method is most likely to produceworstresult. Since the empirical relationship between risk and reward has always been negative, we might expect optimal optimizations to produce worst-case results when the relationship is positive. Optimizing maximum diversification is particularly optimal when returns are proportional to volatility. It makes sense that this portfolio would lag the performance of a portfolio of equal weight and minimum variance assuming no relationship.
Asset Class Portfolio
Stocks and bonds have equal historical Sharpe ratios when the performance of the four economic regimes described by the combination of inflation and growth shocks is averaged.6The historical Sharpe ratio for commodities is about half that for stocks and bonds. However, since our sample size includes only a small number of institutions dating back to 1970, we are reluctant to reject the practical assumption that true Sharpe ratios for diversified commodity portfolios are consistent with those for stocks and bonds.
Assuming that stocks, bonds, and commodities have similar Sharpe ratios, the decision tree approach of the optimization machine can find the optimal mean-variance portfolio using maximum diversification optimization. A risk parity portfolio should also work well, since it is optimal when the asset's marginal Sharpe ratio equals that of the equal risk contribution portfolio.
Minimum variance and equally weighted portfolios are likely to return the weaker Sharpe indices because their associated optimal conditions are more likely to be violated. Major asset classes are generally uncorrelated, while subcategories (ie regional indices) are more highly correlated, so the universe should have heterogeneous correlations. Also, bonds should be much less volatile than other assets. In the end, the returns on individual assets should be far from equal, as riskier assets should have higher returns.
With our assumptions in mind, let's check the results of our simulations.
Simulation results
We run simulations for each target investment domain to compare the simulated performance of portfolios formed using naive and optimization-based approaches. In cases where optimization requires volatility or covariance estimates, we use the last 252 days to form our estimates. We do not shrink apart from restricting the portfolio to long positions only with weights that add up to 100%. The portfolio is rebalanced quarterly.
For illustrative purposes, Figure 5 shows $1 growth to simulate the range of our 25 portfolios sorted by price and market value. Based on out-of-the-box leverage, and for comparison, we adjust each portfolio to the same ex-post volatility as the capitalization-weighted portfolio.7We assume that the annual cost of leverage is equal to the 3-month Treasury bill rate plus one percent. Scaled at equal volatilities, the portfolio formed using the minimum variance produced the best performance between 1927 and 2017.
Figure 5:Naive vs. Robust Portfolio Optimization $1 Growth, 25 Factor Portfolios Ranked by Size and Market Share, 1927-2018
Source: ReSolve Asset Management. Simulation results. Portfolios are formed based on industry returns, factor portfolios, and monthly asset class returns over the past 252 days. The result is the total amount of costs related to the transaction. For illustrative purposes only.
Table 1 summarizes the Sharpe indices for each optimization method applied to each universe.
Tabla 1:Performance Statistics: Naive vs. Robust Portfolio Optimization. Industry and factor simulations from 1927 to 2017. Asset class simulation from 1990 to 2017.
| 10 industries | 25 factor classifications | 38 industries | 49 industries | 12 asset classes | |
|---|---|---|---|---|---|
| Sharpe ratio of average assets | 0.44 | 0,46 | 0.37 | 0.37 | |
| cap weight | 0.44 | 0.44 | 0.44 | 0.44 | 0,68 |
| equal weight | 0.50 | 0.51 | 0.49 | 0.50 | 0.56 |
| minimum variance | 0,53 | 0,62 | 0.58 | 0,55 | 0,61 |
| maximum diversification | 0.50 | 0.41 | 0,55 | 0,53 | 0.76 |
| Hierarchical Minimum Variance | 0.51 | 0.56 | 0,55 | 0,55 | 0,66 |
| contribution of equal risk | 0.51 | 0.54 | 0.54 | 0.54 | 0.71 |
| reverse volatility | 0.51 | 0.54 | 0.52 | 0.52 | 0,69 |
| Optimize Combination | 0.52 | 0,55 | 0.57 | 0.56 | 0.72 |
Source: ReSolve Asset Management. Simulation results. Portfolios are formed based on industry returns, factor portfolios, and monthly asset class returns over the past 252 days. The result is the total amount of costs related to the transaction. For illustrative purposes only.
Discussion of simulation results
stock
The first thing to note is that all of the methods outperform market capitalization-weighted portfolios, with a few notable exceptions: the most diversified portfolio underperforms market capitalization-weighted portfolios in the ranking domain. of factors.
This is no surprise. Market capitalization-weighted portfolios are optimal in terms of mean variance if stock returns can be explained by them.bto the market, making the stock have a higherbThere is a correspondingly higher return. However, in ourPortfolio Optimization White PaperInvestors are not adequately compensated for taking on additional market riskbTherefore, investors in market capitalization-weighted portfolios assume additional risks for which they are not compensated.
The evidence confirms our hypothesis that minimum variance portfolios should generate the best risk-adjusted returns in equity-oriented sectors. The machine's decision tree optimization also shows that the maximum diversification strategy should perform worse in the stock sector due to the flat (or even negative) empirical relationship between stock risk and return. In fact, maximum diversification lags behind other optimizations in some simulations. However, it outperformed the inverse volatility and equal-risk contribution methods in many cases, and it dominated equal-weighted portfolios in industry simulations38 and 49.
asset class
We believe that diversified asset classes should have equal long-term Sharpe ratios, which leads us to assume that the most diversified portfolios should dominate the asset class universe. We expect equal weight, minimum variance strategies to underperform. These predictions were played out in the simulations. Equal risk contributions and inverse volatility weighting approaches are also competitive, suggesting that the assumption of constant correlation may not be too far off target.
In addition to publishing results for each portfolio selection method, we also publish results for a portfolio that averages the weights of all optimization strategies for each epoch. For all universes except the factor-ordered universe, the unbiased average of all optimizations (including the least optimal strategy) outperforms the naive equal-weighted approach.
significance test
Although methods based on proper optimization produce better results than weighting each universe equally, it is useful to check whether the results are statistically significant. After all, the performance gains seen with the best optimization methods aren't huge.
To determine if the results are economically significant or simply an artifact of randomness, we perform a block bootstrap test for the Sharpe relation. Specifically, we randomly sample four quarterly segments of returns (with 12 monthly returns across the entire range of asset classes) and apply replacement to create 10,000 potential return streams for each strategy.
Each draw consists of an equally weighted sample of returns, as well as the returns of the target optimal strategy, with the same random date index. Each sample is the same length as the original simulation.
We then compare the Sharpe indices of each sample of the equally weighted returns with the Sharpe indices of the samples of the optimally weighted returns. The values in Table 2 represent the proportion of samples for which the equal-weighted return sample Sharpe index exceeds the optimal strategy return sample Sharpe index. Thus, they are similar to traditional p-values, where p is the probability that the best strategy will outperform due to chance.
Tabla 2:The pairwise probability that the Sharpe ratio of the optimization-based strategy is less than or equal to the Sharpe ratio of the equal weighting strategy.
| 10 industries | 25 factor classifications | 38 industries | 49 industries | 12 asset classes | |
|---|---|---|---|---|---|
| minimum variance | 0.07 | 1.00 | 0.01 | 0.44 | 0.01 |
| maximum diversification | 0.19 | 1.00 | 0.26 | 0,94 | 0.01 |
| Hierarchical Minimum Variance | 0.01 | 0.00 | 0.00 | 0.00 | 0.07 |
| contribution of equal risk | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| reverse volatility | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| Optimize Combination | 0.00 | 0.97 | 0.00 | 0.08 | 0.00 |
Source: ReSolve Asset Management. Simulation results. Portfolios are formed based on industry returns, factor portfolios, and monthly asset class returns over the past 252 days. The result is the total amount of costs related to the transaction. For illustrative purposes only.
The analysis yielded some surprising results. While the minimum variance strategy produces the highest sample Sharpe indices for all equity-oriented domains, risk parity-based approaches such as equal risk contributions and inverse volatility are still more advantageous from a statistical point of view. The hierarchical minimum variance method also exhibits a high degree of statistical robustness.
For the asset class domain, all portfolios outperform equal-weighted portfolios on a statistically significant basis, except the stratified minimum variance portfolio. The stratified minimum variance portfolio outperformed the equal-weighted portfolio 93% of the time. This further validates the importance of optimization when asset ranges have different volatility and correlation characteristics.
Risk parity approaches are more likely to dominate equal-weighted portfolios, as they exhibit less active riskRelative to an Equally Weighted Portfolio.However, while risk parity portfolios may outperform equal-weighted portfolios a bit more often on a relative basis, they may more oftenlowPerforms minimum variance and maximum spread on an absolute basis for stock universes and asset classes, respectively.
Summary and next steps
Many investment professionals mistakenly believe that portfolio optimization is too noisy to be useful in practice. This myth stems from a series of widely cited articles that purport to prove that portfolio optimization cannot outperform naive methods.
The purpose of this article is to illustrate how portfolio optimization machines can be a useful framework for determining which optimization method is best for a given investment domain. We use optimization machines in conjunction with data and beliefs to formulate hypotheses about optimal portfolio choices in various investment domains. We then proceed to test the hypothesis through the simulation results on real-time data.
Selecting portfolio optimizer calls produced excellent results. Both naive and optimal approaches dominate market capitalization-weighted portfolios. In most cases, optimization-based methods dominate naive equal-weighting methods, unless the optimization expresses a risk-reward relationship that is the exact opposite of what has been observed historically. For example, maximum diversification indicates a positive relationship between returns and volatility, while stocks have historically shown a flat or even inverted relationship. Therefore, it should come as no surprise to learn that highly diversified portfolios underperform equally weighted portfolios when applied in certain equity-oriented domains.
While optimization-based methods compete with the performance of naive methods in the cases studied in this paper, we recognize that our test cases may not be representative of the real-world challenges faced by many portfolio managers. Many portfolio selection problems involve a large number of securities with high average correlations. Investors also often require limits on sector risk, tracking error, factor exposures, and portfolio concentration. Other investors may be long/short portfolios, which introduces a higher degree of instability.
We sympathize with the fact that most financial professionals are not trained in numerical methods. While portfolio optimization is covered in the CFA and most MBA courses, the topic is limited to the most basic two-asset case of traditional mean-variance optimization with known means and covariances. Of course, this is quite different from real world portfolio selection problems.
In future articles, we'll explore more challenging problems involving low-quality investment domains with more typical constraints. We will delve into some of the mathematical challenges posed by optimization and propose novel solutions supported by robust simulations. Later, we describe how to incorporate a dynamic active view of asset returns informed by systemic factors, which we refer to as "Adaptive Asset Allocation”
reference
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Choueifaty, Yves and Yves Coignard. 2008. "Towards Maximum Diversity".Portfolio Management Magazine35 (1)http://www.tobam.fr/inc/uploads/2014/12/TOBAM-JoPM-Maximum-Div-2008.pdf：40-51。
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FAQs
What are the 2 methods of optimizing portfolio? ›
Portfolio optimization often takes place in two stages: optimizing weights of asset classes to hold, and optimizing weights of assets within the same asset class.
What are the best models for portfolio optimization? ›The most popular method that does incorporate views is the Markovitz Mean-Variance Optimal portfolio based on the Capital Asset Pricing Model or CAPM. The passive portfolios like the market index use a market-cap-weighted allocation.
How would you determine the optimal portfolio among the efficient set of risky assets? ›The optimal risky portfolio is found at the point where the CAL is tangent to the efficient frontier. This asset weight combination gives the best risk-to-reward ratio, as it has the highest slope for CAL.
What is the difference between efficient portfolio and optimal portfolio? ›Efficient portfolio is a portfolio that offers the greatest expected return with certain risks or offers the lowest level of risk with a certain expected return. Optimal portfolio is the portfolio that gives the optimal expected return with the smallest risk from the arrangement of investor choices [5].
What are the two 2 methods of analyzing investments? ›Fundamental vs.
Other investment analysis methods include fundamental analysis and technical analysis. The fundamental analyst stresses the financial health of companies as well as the broader economic outlook. Practitioners of fundamental analysis seek stocks they believe the market has mispriced.
It involves clarifying, prioritizing, and selecting the projects that are best aligned with the overall business objectives of the firm and determining the optimal way to sequence timelines in order to make the most out of the enterprise's project activity.
Which portfolio strategy is best? ›- Invest in Alternative Assets Like Fine Wine.
- Invest in Dividends.
- Invest in Non-Correlating Assets.
- Invest in Principal-Protected Notes.
- Diversify Your Portfolio.
- Buy Put Options.
- Use Stop-Loss Orders.
- Find a Financial Advisor.
The BCG analysis was developed by the Boston Consulting Group and is one of the best known portfolio analyses. It is also known as BCG Matrix.
What is the best known portfolio planning method? ›The Boston Consulting Group (BCG) Matrix is the best-known approach to portfolio planning (Table 8.5). Using the matrix requires a firm's businesses to be categorized as high or low along two dimensions: its share of the market and the growth rate of its industry.
What is the formula for optimal risk in a portfolio? ›The optimal proportion of the risky asset in the complete portfolio is given by the equation y * = [E(rP)-rf]/(. 01A * Variance of P).
How do I know which portfolio is riskier? ›
The most fundamental thing to understand is that the proportion of a portfolio that goes into equities is the key factor in determining its risk profile. Most sources cite a low-risk portfolio as being made up of 15-40% equities. Medium risk ranges from 40-60%. High risk is generally from 70% upwards.
How do you identify an optimal risky portfolio? ›Answer and Explanation: The optimal risky portfolio can be identified by finding a point of the capital market line that is tangent to the efficient frontier and the line with the steepest slope that connects the risk-free rate to the efficient frontier.
How many stocks are considered optimal in a portfolio? ›Generally speaking, many sources say 20 to 30 stocks is an ideal range for most portfolios. It's important to strike a balance between investing in a diverse array of assets and ensuring that you have the time and resources to manage these investments.
What is the difference between efficient and inefficient portfolio? ›An inefficient portfolio is one that delivers an expected return that is too low for the amount of risk taken on. Conversely, an inefficient portfolio also refers to one that requires too much risk for a given expected return. In general, an inefficient portfolio has a poor risk-to-reward ratio.
What is optimal portfolio in simple words? ›An optimal portfolio is one designed with a perfect balance of risk and return. The optimal portfolio looks to balance securities that offer the greatest possible returns with acceptable risk or the securities with the lowest risk given a certain return.
What 3 factors do investment analysis methods generally evaluate? ›Investment analysis methods generally evaluate 3 factors: risk, cash flows, and resale value.
What are two 2 ways investors profit from stocks and mutual funds? ›- Dividend Payments. A fund may earn income from dividends on stock or interest on bonds. ...
- Capital Gains Distributions. The price of the securities in a fund may increase. ...
- Increased NAV.
There are two primary methods for analyzing financial statements. One is horizontal and vertical analysis, where horizontal analysis compares data sets across certain time periods, while vertical analysis reports costs and assets as a percentage of the entire financial statement.
What are the 4 Ps of portfolio management? ›These are People, Philosophy, Process, and Performance. When evaluating a wealth manager, these are the key areas to think about. The 4P's can be dissected further, but for the purpose of this introduction, we'll focus on these high-level categories.
What are the 5 phases of portfolio management? ›- Step 1 – Identification of objectives. ...
- Step 2 – Estimating the capital market. ...
- Step 3 – Decisions about asset allocation. ...
- Step 4 – Formulating suitable portfolio strategies. ...
- Step 5 – Selecting of profitable investment and securities. ...
- Step 6 – Implementing portfolio. ...
- Step 7 – ...
- Step 8 –
What is the best mix of portfolio? ›
Income Portfolio: 70% to 100% in bonds. Balanced Portfolio: 40% to 60% in stocks. Growth Portfolio: 70% to 100% in stocks. For long-term retirement investors, a growth portfolio is generally recommended.
What is the most efficient portfolio? ›1. The market portfolio is an efficient portfolio: its allocation provides the only optimal mix of risky assets; 2. For each asset, its expected return follows a simple linear relationship with the expected return of the market portfolio.
What is the most simple investment strategy? ›- Buy and hold. A buy-and-hold strategy is a classic that's proven itself over and over. ...
- Buy index funds. This strategy is all about finding an attractive stock index and then buying an index fund based on it. ...
- Index and a few. ...
- Income investing. ...
- Dollar-cost averaging.
A mutual fund or index fund provides more diversification than an individual security does. It tracks a bundle of stocks, bonds, or commodities.
What is the best portfolio diversification? ›A diversified portfolio should have a broad mix of investments. For years, many financial advisors recommended building a 60/40 portfolio, allocating 60% of capital to stocks and 40% to fixed-income investments such as bonds.
What are the three main criteria used for portfolio analysis? ›There are three key criteria to take into consideration when assessing a portfolio: the combined value of the projects in the portfolio, the overall risk/value balance, and the alignment of the portfolio with the strategic goals of your business.
What are the 2 main types of portfolio? ›There are two main types of portfolio assessments: “instructional” or “working” portfolios, and “showcase” portfolios. Instructional or working portfolios are formative in nature. They allow a student to demonstrate his or her ability to perform a particular skill. Showcase portfolios are summative in nature.
What is the 1% rule for managing risk? ›One of the most popular risk management techniques is the 1% risk rule. This rule means that you must never risk more than 1% of your account value on a single trade. You can use all your capital or more (via MTF) on a trade but you must take steps to prevent losses of more than 1% in one trade.
How do you create an optimal portfolio? ›To create an Optimal Portfolio one of the main aspects is Risk Diversification. It can be achieved by using some technical ideologies. Optimal portfolio is a term used to refer Efficient Frontier with the highest return-to-risk combination given the specific investor's tolerance for risk.
How do you calculate portfolio optimization? ›Explanation. An optimal portfolio is said to have the highest Sharpe ratio. You can calculate it by, Sharpe Ratio = {(Average Investment Rate of Return – Risk-Free Rate)/Standard Deviation of Investment Return} read more, which measures the excess return generated for every unit of risk taken.
Which portfolio has the least risk? ›
- Short-term certificates of deposit. ...
- Money market funds. ...
- Treasury bills, notes, bonds and TIPS. ...
- Corporate bonds. ...
- Dividend-paying stocks. ...
- Preferred stocks. ...
- Money market accounts. ...
- Fixed annuities.
The greatest risk facing any portfolio is market risk. This is also known as systematic risk.
What are the two types portfolios risk can be broken down into? ›Risk and Return: The Portfolio Theory
The risk (variance) on any individual investment can be broken down into two sources: - Firm specific risk (only faced by that firm), - Market wide risk (affects all investments).
This sort of five percent rule is a yardstick to help investors with diversification and risk management. Using this strategy, no more than 1/20th of an investor's portfolio would be tied to any single security. This protects against material losses should that single company perform poorly or become insolvent.
How many stocks should I own with $100 K? ›A good range for how many stocks to own is 15 to 20. You can keep adding to your holdings and also invest in other types of assets such as bonds, REITs, and ETFs. The key is to conduct the necessary research on each investment to make sure you know what you are buying and why.
How much is too many stocks in a portfolio? ›Depending on which research you pull, you can find arguments suggesting that anywhere between 10 and 60 individual stocks will make up a well-diversified series of investments. However, for investors looking for a rule of thumb, we would suggest considering this from a budget-first perspective: Invest with funds.
Are efficient portfolios always optimal? ›The Efficient Frontier is a set of optimal portfolios that give the highest possible expected return for a given risk level or the lowest risk for a desired expected return. Portfolios below the efficient frontier are sub-optimal, as they don't provide enough returns for their risk levels.
What are efficient portfolio management techniques? ›Efficient Portfolio Management or “EPM” means investment in Derivatives with the aim of reducing risk or costs for the Fund or with the aim of generating additional Capital or Income without any additional risk. A common example of the use of EPM is Hedging in order to reduce risk.
What are the main characteristics of an efficient portfolio? ›- Risk averse. Your portfolio should not expose you to any more risk than is necessary to meet your objectives. ...
- Cost efficient. A good portfolio achieves its objectives at the lowest possible cost. ...
- Risk efficient. ...
- Tax efficient. ...
- Simple. ...
- Transparent. ...
- Easy to manage.
Efficient portfolio is a portfolio that offers the greatest expected return with certain risks or offers the lowest level of risk with a certain expected return. Optimal portfolio is the portfolio that gives the optimal expected return with the smallest risk from the arrangement of investor choices [5].
How does an investor select an optimal portfolio? ›
According to the mean-variance criterion, any investor would optimally select a portfolio on the upward-sloping portion of the portfolio frontier, which is called the efficient frontier, or minimum variance frontier. The choice of any portfolio on the efficient frontier depends on the investor's risk preferences.
What are the two types of portfolio theory? ›In the case of using Formula plans, the amount to invest is known, and two types of portfolios are made. One portfolio is aggressive in nature that invests in risky assets and stocks. The other portfolio is of defensive nature, and investments are made in safe assets such as government bonds, debentures, etc.
How do you optimize your portfolio? ›When optimizing your portfolio, you assign an 'optimization weight' for each asset class and all assets within that class. The weight is the percentage of the portfolio that concentrates within any particular class. For example, say we weigh stocks at 10% and bonds at 20%.
What are the 4 types of portfolio planning? ›- Active Portfolio Management. The aim of the active portfolio manager is to make better returns than what the market dictates. ...
- Passive Portfolio Management. ...
- Discretionary Portfolio Management. ...
- Non-Discretionary Portfolio Management.
- Conservative portfolio. This type is also called a defensive portfolio or a capital preservation portfolio. ...
- Aggressive portfolio. Also known as a capital appreciation portfolio. ...
- Income portfolio. ...
- Socially responsible portfolio.
- Portfolio governance.
- Portfolio management.
- Demand management.
- Resource management.
- Cost management.
- Outcome/Benefits management.
- Schedule management & roadmaps.
- Program management.
An optimal portfolio is said to have the highest Sharpe ratio. You can calculate it by, Sharpe Ratio = {(Average Investment Rate of Return – Risk-Free Rate)/Standard Deviation of Investment Return} read more, which measures the excess return generated for every unit of risk taken.
What are the 5 types of portfolio? ›- The Aggressive Portfolio.
- The Defensive Portfolio.
- The Income Portfolio.
- The Speculative Portfolio.
- The Hybrid Portfolio.
Accelerator Theory Of Investment, Internal Funds Theory Of Investment, and Neoclassical Theory Of Investment are three major types of investment theories. These theories can be used by representative parties to establish their views on the nature of the financial markets and make decisions to reach their broad goals.
What is the basic portfolio theory? ›The modern portfolio theory (MPT) was a breakthrough in personal investing. It suggests that a conservative investor can do better by choosing a mix of low-risk and riskier investments than by going entirely with low-risk choices.
What is the new portfolio theory? ›
Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk.
What is an example of portfolio optimization? ›In investing, portfolio optimization is the task of selecting assets such that the return on investment is maximized while the risk is minimized. For example, an investor may be interested in selecting five stocks from a list of 20 to ensure they make the most money possible.
What is portfolio optimization model? ›Portfolio optimization is a formal mathematical approach to making investment decisions across a collection of financial instruments or assets. Portfolios are points from a feasible set of assets that constitute an asset universe.
What is the most optimal portfolio? ›An optimal portfolio is one designed with a perfect balance of risk and return. The optimal portfolio looks to balance securities that offer the greatest possible returns with acceptable risk or the securities with the lowest risk given a certain return.